The minimum value of \(Z = 2x + 5y\) subject to the constraints \(x + y = 6\), \(x \le 4\), \(x \ge 0\), \(y \ge 0\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 18
Because of the equation \(x + y = 6\), every feasible point lies on this line. With \(0 \le x \le 4\) and \(y = 6 - x \ge 0\), the feasible region is the line segment joining
\((0, 6)\) and \((4, 2)\)
Its end points are the corner points. Values of Z:
\(Z(0, 6) = 30\)
\(Z(4, 2) = 8 + 10 = 18\)
So the minimum value is 18, at (4, 2).
Why the other options are incorrect
Option (1) 12 — Incorrect. This is Z at (6, 0), which breaks \(x \le 4\).
Option (2) 15 — Incorrect. This is Z at (5, 1), which also breaks \(x \le 4\).
Option (4) 30 — Incorrect. This is the maximum value, at (0, 6).